Revisiting Fracture Mechanic Handbooks
with (Interpretable) Sparse Regression

F. Loiseau (IMSIA)
Y. Epongue Djeugoue, V. Lazarus (IMSIA)
G. Eliyahu-Yakir, P. Reis (EPFL).

MePhy & I-Gaia Day: Machine Learning in mechanics and physics
December 9, 2025.

Linear Elastic Fracture Mechanics

The S.S. Schenectady tanker split apart by brittle fracture while in harbor, 1943.
Source : https://en.wikipedia.org/wiki/Fracture_mechanics.

Typical problem

An elastic body \(\Omega\) with a crack \(\Gamma\) under a given load.

Assumption : Non-linearities are confined close to the crack tip.

Question : Does the crack propagates ?

Crack tip singularity

Stress field (Williams, 1957)

\[ \sigma_{ij}(r, \theta) = \frac{\color{red}{K_I}}{\sqrt{2\pi r}} f_{ij}^{I}(\theta) + \frac{\color{red}{K_{II}}}{\sqrt{2\pi r}} f_{ij}^{II}(\theta) + \frac{\color{red}{K_{III}}}{\sqrt{2\pi r}} f_{ij}^{III}(\theta) + \mathcal{O}(1), \quad \color{gray}{\sigma_{ij}(r, \theta) \underset{r \to 0}{\to} \infty.} \]

There is a stress singularity at crack tip.




Stress Intensity Factors (SIFs)

Denoted \(K_p\), they :

  • represent the crack tip sollicitation,
  • depend on geometry and load.

Designing structures with cracks

Crack propagation criteria

Propagation criteria rely on the SIFs \(K_p\), e.g.,

\(\quad \displaystyle \color{red}{K_I} \leq K_{Ic}\), or

\(\quad \displaystyle G = \frac{\color{red}{K_{I}}^2 + \color{red}{K_{II}}^2}{E'} + \frac{\color{red}{K_{III}}^2}{2 \mu} \leq G_c\).

How to determine the SIFs ?

Different methods are employed :

  • analytical (e.g., crack in (semi-)infinite domain TODO: Change to a specific method),
  • numerical (e.g., FEM simulations),
  • experimental (e.g., photo-elasticity, DIC.).

Known solutions have been compiled in handbooks!

Fracture mechanic design handbooks
(Tada et al., 2000)

Example (Tada et al., 2000)

Methodology

  1. Dimensional analysis to find relevant parameters
    • dimensionless function \(F(\color{green}{\mathbf{p}})\),
      depending on parameters \(\color{green}{\mathbf{p}}\) (geometry & load).
  2. Find \(F(\color{green}{\mathbf{p}})\) as truncated series expansion, or
    fit of numerical/experimental data, or …

Limitations

Dense polynomials, limited to few parameters, etc.

Finding a sparse analytical model

Colors: \(\color{green}{\text{Inputs (parameters)} \color{green}{\mathbf{p}}}, \color{blue}{\text{Coefficients } \color{blue}{\mathbf{c}}}, \color{red}{\text{Outputs } \color{red}{K_{p}}}\).

Objective

For a structure and load parameterized by \(\color{green}{\mathbf{p}}= [\color{green}{p_{1}}, \color{green}{p_{2}}, ..., \color{green}{p_{N}}]^{T}\), find the relationship \(\color{red}{K_{p}}(\color{green}{\mathbf{p}})\).

Tool : LASSO regression (Tibshirani, 1996)

We look for a polynomial approximation of \[ \color{red}{K_{p}}(\color{green}{\mathbf{p}}) \approx \color{red}{\widetilde{K}_{p}^{\color{blue}{\mathbf{c}}}}(\color{green}{\mathbf{p}}) = \sum_{i_1} \sum_{i_2} \cdot\cdot\cdot \sum_{i_N} \color{blue}{c_{i_1, i_2, ..., i_n}} \color{green}{p_{1}}^{i_1} \color{green}{p_{2}}^{i_2} ... \color{green}{p_{N}}^{i_P}. \] For \(N_{\mathrm{obs}}\) observations (\(\color{green}{\mathbf{p}}\), \(\color{red}{K_{p}}\)), the optimal coefficients \(\color{blue}{\mathbf{c}^*}\) minimizes \[ \color{blue}{\mathbf{c}^*} = \arg\min_{\color{blue}{\mathbf{c}}} \frac{1}{2 N_{\mathrm{obs}}} \sum_{\boldsymbol{p}} \left(\color{red}{\widetilde{K}_{p}^{\color{blue}{\mathbf{c}}}}(\color{green}{\mathbf{p}}) - \color{red}{K_{p}}(\color{green}{\mathbf{p}}) \right)^2 + \beta \color{blue}{\| \color{blue}{\mathbf{c}}\|_1} , \quad \color{gray}{\| \boldsymbol{c} \|_1 = \sum_n | c_n |.} \] The L1-norm of the coefficients \(\color{blue}{\mathbf{c}}\) drives irrelevant coefficients to zero.

Adding non-polynomial terms to model

Replace \(\color{green}{\mathbf{p}}\) with \(\mathbf{s} = [\color{green}{p_{1}}, f(\color{green}{p_{1}}), \color{green}{p_{2}}, f(\color{green}{p_{2}}), ..., \color{green}{p_{N}}, f(\color{green}{p_{N}})]\).

Ex. 1 : CT specimen - Setup & Data

Compact Tension (CT) specimen

Sparse regression setup

Observations \(a\), \(K_I\) (from numerical simulation)
Parameters \(a\), \(\exp(a)\), \(\frac{1}{\sqrt{b-a}}\)
Output \(\hat{K}_I = \frac{K_I}{\sigma \sqrt{a}}\) with \(\sigma=P/b\)


Observations

Ex. 1 : CT specimen - Tradeoff - Deg=2

Code output

=== Model 1
Assessment :
    - Hyperparameter beta    : 3.83e+02
    - Number of coefficients : 1
    - R2 score               : 0.000
LaTeX expression:
\hat{K}_{I}(a) =
    +45.6

=== Model 2
Assessment :
    - Hyperparameter beta    : 1.02e+01
    - Number of coefficients : 2
    - R2 score               : 0.967
LaTeX expression:
\hat{K}_{I}(a) =
    -21.1
    +17.9 \frac{1}{\sqrt{b-a}}^2

=== Model 3
Assessment :
    - Hyperparameter beta    : 5.11e-01
    - Number of coefficients : 3
    - R2 score               : 0.997
LaTeX expression:
\hat{K}_{I}(a) =
    +5.43
    -14.4 \exp(a/b)^2
    +22.9 \frac{1}{\sqrt{b-a}}^2

Ex. 1 : CT specimen - Tradeoff - Deg=3

Code output

=== Model 1
Assessment :
    - Hyperparameter beta    : 1.79e+03
    - Number of coefficients : 1
    - R2 score               : 0.000
LaTeX expression:
\hat{K}_{I}(a) =
    +45.6

=== Model 2
Assessment :
    - Hyperparameter beta    : 3.43e+00
    - Number of coefficients : 2
    - R2 score               : 0.999
LaTeX expression:
\hat{K}_{I}(a) =
    +3.75
    +4.11 \frac{1}{\sqrt{b-a}}^3

=== Model 3
Assessment :
    - Hyperparameter beta    : 6.13e-01
    - Number of coefficients : 3
    - R2 score               : 1.000
LaTeX expression:
\hat{K}_{I}(a) =
    +5.32
    -0.383 \exp(a/b)^3
    +4.19 \frac{1}{\sqrt{b-a}}^3

Ex. 2 : Rotating structure - Setup & Data


Experimental setup

By G. Eliyahu-Yakir and P. Reis (EPFL)

Plate geometry

Illustration

Ex. 2 : Rotating structure - Physics

Illustration

7-parameter problem

  • Geometry: \(\color{green}{W, L, a, L_c}\).
  • 3D volumic loads:
    • centrifugal : \(\boldsymbol{f}_\Omega(r, \theta) \propto \color{green}{\Omega}\)
      • along \(\boldsymbol{e}_r\) and \(\boldsymbol{e}_\theta\),
    • Euler : \(\boldsymbol{f}_\dot{\Omega}(r, \theta) \propto \color{green}{\dot{\Omega}}\)
      • along \(\boldsymbol{e}_2 = \cos(\color{green}{\alpha}) \boldsymbol{e}_\theta - \sin(\color{green}{\alpha}) \boldsymbol{e}_z\).

Dimensional analysis (towards a 3-parameter problem)

The superposition principle (\(K_p = K_{p}^{\Omega} + K_{p}^{\dot{\Omega}}\)) and the Buckingham \(\Pi\) theorem give (\(\frac{L}{W}\) is fixed)

  • centrifugal load
    • \(K_{I}^{\Omega} = \rho \color{green}{\Omega}^2 \color{green}{W}^{5/2} \color{red}{\hat{K}_{I}^{\Omega}}\left(\color{green}{\frac{a}{W}}, \color{green}{\frac{L_c}{L}}, \color{gray}{\frac{L}{W}}\right)\)
    • \(K_{II}^{\Omega} = \rho \color{green}{\Omega}^2 \color{green}{W}^{5/2} \color{red}{\hat{K}_{II}^{\Omega}}\left(\color{green}{\frac{a}{W}}, \color{green}{\frac{L_c}{L}}, \color{gray}{\frac{L}{W}}\right)\)
    • \(K_{III}^{\Omega} = 0\),
  • Euler load
    • \(K_{I}^{\dot{\Omega}} = \rho \color{green}{\dot{\Omega}} \color{green}{W}^{5/2} \cos(\color{green}{\alpha}) \color{red}{\hat{K}_{I}^{\dot{\Omega}}}\left(\color{green}{\frac{a}{W}}, \color{green}{\frac{L_c}{L}}, \color{gray}{\frac{L}{W}}\right)\)
    • \(K_{II}^{\dot{\Omega}} = \rho \color{green}{\dot{\Omega}} \color{green}{W}^{5/2} \cos(\color{green}{\alpha}) \color{red}{\hat{K}_{II}^{\dot{\Omega}}}\left(\color{green}{\frac{a}{W}}, \color{green}{\frac{L_c}{L}}, \color{gray}{\frac{L}{W}}\right)\)
    • \(K_{III}^{\dot{\Omega}} = \rho \color{green}{\dot{\Omega}} \color{green}{W}^{5/2} \sin(\color{green}{\alpha}) \color{red}{\hat{K}_{III}^{\dot{\Omega}}}\left(\color{green}{\frac{a}{W}}, \color{green}{\frac{L_c}{L}}, \color{gray}{\frac{L}{W}}\right)\).

Ex. 2 : Rotating structure - Data

Data generation

Elastic FEM simulations

Ex. 2 : Rotating structure - Model \(\hat{K}_{I}^{\dot{\Omega}}\)

Ex. 2 : Rotating structure - All the SIFs

Model \(N_{\mathrm{coeff}}\) \(R^2\)
\(\displaystyle \hat{K}_{I}^{\Omega} = \exp\left( 2.77 + 2.72 \exp\left(\frac{a}{W}\right) - 0.986 \exp\left(\frac{Lc}{L}\right) \right)\) 3 0.995
\(\displaystyle \hat{K}_{II}^{\Omega} = 0\) - -
\(\displaystyle \hat{K}_{III}^{\Omega} = 0\) -
\(\displaystyle \hat{K}_{I}^{\dot{\Omega}} = \exp \left(8.25 + 3.25 \left(\frac{a}{W}\right)^2 - 3.9 \left(\frac{L_c}{L}\right)^2 \right)\) 3 0.996
\(\displaystyle \hat{K}_{II}^{\dot{\Omega}} = -5.75 -127 \left(\frac{a}{W}\right)^2 -394 \frac{a}{W} \left(1-\frac{Lc}{L}\right)\) 3 0.989
\(\displaystyle \hat{K}_{III}^{\dot{\Omega}} = 52.7 + 238 \left(\frac{a}{W}\right) \left(1-\frac{Lc}{L}\right) + 97.4 \left(\frac{a}{W}\right)^3 - 66.6 \left(\frac{L_c}{L}\right)^3\) 4 0.995

The fracture mechanics problem is now fully analytical.

Conclusion


Summary of the methodology

  1. Physics : Dimensional analysis + Superposition
    • Isolate independent SIFs
    • Find the proper dimensionless parameters
  2. Generation of the data
    • Sampling of the parametric space (LHS)
    • Elastic simulations
  3. Machine learning : Sparse regression
    • Large polynomial + L1-regularization
    • Injection of non-linear terms as model input
  4. Result : Sparse analytical approximation of the SIFs



Benefits in fracture mechanics

A problem of fracture can now be fully analytical facilitating

  • uncertainty quantification,
  • design (parametric optimization of structure),
  • stability analysis,
  • etc.

Thanks for your attention !

F. Loiseau, Y. Epongue Djeugoue, G. Eliyahu-Yakir, P. Reis, V. Lazarus.

flavien.loiseau@ensta.fr

Presentation



Ce travail est financé par l’Agence de l’Innovation de Défense – AID – via le Centre Interdisciplinaire d’Etudes pour la Défense et la Sécurité – CIEDS – (projects 2022 - FracAddi).

Références

Tada, H., Paris, P. C., & Irwin, G. R. (2000). The Stress Analysis of Cracks Handbook, Third Edition. ASME Press. https://doi.org/10.1115/1.801535
Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society. Series B (Methodological), 58(1), 267–288. https://www.jstor.org/stable/2346178
Williams, M. L. (1957). On the Stress Distribution at the Base of a Stationary Crack. Journal of Applied Mechanics, 24(1), 109–114. https://doi.org/10.1115/1.4011454